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Let O be a Dedekind domain whose field of fractions K is a global field. Let A be a finite-dimensional separable K -algebra and let Λ be an O -order in A. Suppose that X is a Λ-lattice such that K ⊗ O X is free of some finite rank n over A. Then X contains a (non-unique) free Λ-sublattice of rank n. The main result of the present article is to show there exists such a sublattice Y such that the generalised module index [ X : Y ] O has explicit upper bounds with respect to division that are independent of X and can be chosen to satisfy certain conditions. We give examples of applications to the approximation of normal integral bases and strong Minkowski units, and to the Galois module structure of rational points over abelian varieties. [ABSTRACT FROM AUTHOR]

We prove the first unconditional consistency result for superpolynomial circuit lower bounds with a relatively strong theory of bounded arithmetic. Namely, we show that the theory V20 is consistent with the conjecture that NEXP⊈P/poly, i.e. some problem that is solvable in non-deterministic exponential time does not have polynomial size circuits. We suggest this is the best currently available evidence for the truth of the conjecture. The same techniques establish the same results with NEXP replaced by the class of problems decidable in non-deterministic barely superpolynomial time such as NTIME(nO(logloglog n)). Additionally, we establish a magnification result on the hardness of proving circuit lower bounds. [ABSTRACT FROM AUTHOR]

We prove that any proof of a ∀ Σ 2 0 sentence in the theory WKL 0 + RT 2 2 can be translated into a proof in RCA 0 at the cost of a polynomial increase in size. In fact, the proof in RCA 0 can be obtained by a polynomial-time algorithm. On the other hand, RT 2 2 has nonelementary speedup over the weaker base theory RCA 0 * for proofs of Σ 1 sentences. We also show that for n ≥ 0 , proofs of Π n + 2 sentences in B Σ n + 1 + exp can be translated into proofs in I Σ n + exp at a polynomial cost in size. Moreover, the Π n + 2 -conservativity of B Σ n + 1 + exp over I Σ n + exp can be proved in PV , a fragment of bounded arithmetic corresponding to polynomial-time computation. For n ≥ 1 , this answers a question of Clote, Hájek and Paris. [ABSTRACT FROM AUTHOR]

In this paper, we investigate weighted spaces of holomorphic functions on some subsets of the complex plane such as the quarter plane and a strip in the upper half plane and we obtain isomorphic classification of these weighted spaces. Also, we characterize the boundedness of composition operators between weighted spaces of holomorphic functions on the quarter plane and strip. At last, we study a special subspace of holomorphic functions on the quarter plane. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and uniqueness of solutions for a class of frictional antiplane contact problems of p(x)-Kirchhoff type on a bounded domain Ω ⊆ R². Using an abstract Lagrange multiplier technique and the Schauder fixed point theorem we establish the existence of weak solutions. Imposing some suitable monotonicity conditions on the datum f1 the uniqueness of the solution is obtained. [ABSTRACT FROM AUTHOR]

Aiming to provide weak as possible axiomatic assumptions in which one can develop basic linear algebra, we give a uniform and integral version of the short propositional proofs for the determinant identities demonstrated overGF (2) in Hrubeš-Tzameret [15]. Specifically, we show that the multiplicativity of the determinant function and the Cayley-Hamilton theorem over the integers are provable in the bounded arithmetic theory VNC²; the latter is a first-order theory corresponding to the complexity class NC² consisting of problems solvable by uniform families of polynomial-size circuits and O(log² n)-depth. This also establishes the existence of uniform polynomial-size propositional proofs operating with NC²-circuits of the basic determinant identities over the integers (previous propositional proofs hold only over the two-element field). [ABSTRACT FROM AUTHOR]

We define and investigate Frege systems for quantified Boolean formulas (QBF). For these new proof systems, we develop a lower bound technique that directly lifts circuit lower bounds for a circuit class C to the QBF Frege system operating with lines from C. Such a direct transfer from circuit to proof complexity lower bounds has often been postulated for propositional systems but had not been formally established in such generality for any proof systems prior to this work. This leads to strong lower bounds for restricted versions of QBF Frege, in particular an exponential lower bound for QBF Frege systems operating with AC0[p] circuits. In contrast, any non-trivial lower bound for propositional AC0[p]-Frege constitutes a major open problem. Improving these lower bounds to unrestricted QBF Frege tightly corresponds to the major problems in circuit complexity and propositional proof complexity. In particular, proving a lower bound for QBF Frege systems operating with arbitrary P/poly circuits is equivalent to either showing a lower bound for P/poly or for propositional extended Frege (which operates with P/poly circuits). We also compare our new QBF Frege systems to standard sequent calculi for QBF and establish a correspondence to intuitionistic bounded arithmetic. [ABSTRACT FROM AUTHOR]

In this paper we prove that the modified K-energy on a Fano manifold X is bounded from below if X admits a special degeneration whose central fiber is a Kähler—Ricci soliton and the soliton vector field can be lifted to X. We will use this result to derive some equivalent criteria of the modified K-semistability. We will also use this result to study the limit of Kähler—Ricci flow. [ABSTRACT FROM AUTHOR]

The process of developing C programs is quite often prone to errors associated with the use of pointer arithmetic and operations on memory addresses. Hence, the need for various automated program verification tools arises. One of the techniques frequently employed by these tools is invocation of suitable decision procedures implemented within existing SMT solvers. However, both the SMT standard and the majority of existing SMT solvers lack relevant logics (combinations of logical theories) for the direct and precise modeling of the semantics of pointer operations in C. A possible way to support these logics is to implement them in an SMT solver. However, this approach can be time-consuming (as it requires modifying the source code of the solver), inflexible (because introducing any changes to the signature or semantics of the theory can be unreasonably hard), and limited (every solver has to be supported individually). Another way is to design and implement custom quantifier instantiation strategies. These strategies can then be used to translate formulas in selected theory combinations into formulas in well-supported decidable logics, e.g., QF_UFLIA. In this paper, we present an instantiation procedure for translating formulas in the theory of bounded pointer arithmetic into the QF_UFLIA logic. We formally prove the soundness and completeness of our instantiation procedure in Isabelle/HOL. The paper also presents an informal description of this proof for the proposed procedure. The theory of bounded pointer arithmetic itself is formulated based on known errors associated with the use of pointer arithmetic operations in C, as well as based on the semantics of these operations specified in the C standard. Similar procedure can also be defined for a practically relevant fragment of the theory of bit vectors (monotone propositional combinations of equalities between bitwise expressions). Our approach is sufficient to derive efficient decision procedures, implemented as Isabelle/HOL proof methods, for several decidable logical theories used in C program verification by relying on the existing capabilities of well-known SMT solvers (e.g., Z3), as well as on the proof reconstruction capabilities of the Isabelle/HOL proof assistant. [ABSTRACT FROM AUTHOR]

We study a new class of NP search problems, those which can be proved total using standard combinatorial reasoning based on approximate counting. Our model for this kind of reasoning is the bounded arithmetic theory APC 2 of [E. Jeřábek, Approximate counting by hashing in bounded arithmetic, J. Symb. Log.  74(3) (2009) 829–860]. In particular, the Ramsey and weak pigeonhole search problems lie in the new class. We give a purely computational characterization of this class and show that, relative to an oracle, it does not contain the problem CPLS, a strengthening of PLS. As CPLS is provably total in the theory T 2 2 , this shows that APC 2 does not prove every ∀ Σ 1 b sentence which is provable in bounded arithmetic. This answers the question posed in [S. Buss, L. A. Kołodziejczyk and N. Thapen, Fragments of approximate counting, J. Symb. Log.  79(2) (2014) 496–525] and represents some progress in the program of separating the levels of the bounded arithmetic hierarchy by low-complexity sentences. Our main technical tool is an extension of the "fixing lemma" from [P. Pudlák and N. Thapen, Random resolution refutations, Comput. Complexity, 28(2) (2019) 185–239], a form of switching lemma, which we use to show that a random partial oracle from a certain distribution will, with high probability, determine an entire computation of a P NP oracle machine. The introduction to the paper is intended to make the statements and context of the results accessible to someone unfamiliar with NP search problems or with bounded arithmetic. [ABSTRACT FROM AUTHOR]

In this paper, we consider a predator-prey model with density-dependent prey-taxis and stage structure for the predator. We establish the existence of classical solutions with uniform-in-time bound in a one-dimensional case. In addition, we prove that the solution stabilizes to the prey-only steady state under some conditions. [ABSTRACT FROM AUTHOR]

We show that V T C 0 , the basic theory of bounded arithmetic corresponding to the complexity class TC 0 , proves the IMUL axiom expressing the totality of iterated multiplication satisfying its recursive definition, by formalizing a suitable version of the TC 0 iterated multiplication algorithm by Hesse, Allender, and Barrington. As a consequence, V T C 0 can also prove the integer division axiom, and (by our previous results) the RSUV -translation of induction and minimization for sharply bounded formulas. Similar consequences hold for the related theories Δ 1 b - C R and C 2 0 . As a side result, we also prove that there is a well-behaved Δ 0 definition of modular powering in I Δ 0 + W P H P (Δ 0) . [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of sign-changing multi-bump solutions for the following nonlinear Choquard equation − (0.1) Δu + (λV(x) + 1)u = (Iα∗|u|p)|u|p−2u in RN, where Iα is the Riesz potential, λ∈R+, (N−4)+ < α < N, 2 ≤ p < (N+α)/(N−2), and V(x) is a nonnegative continuous function with a potential well Ω := int(V−1(0)) which possesses k disjoint bounded components Ω1, …, Ωk . We prove the existence of sign-changing multi-bump solutions for (0.1) if λ is large enough. [ABSTRACT FROM AUTHOR]

In this paper, we study bounded versions of some model-theoretic notions and results. We apply these results to models of bounded arithmetic theories as well as some related complexity questions. As an example, we show that if the theory S2¹(PV) has bounded model companion then NP = coNP. We also study bounded versions of some other related notions such as Stone topology. [ABSTRACT FROM AUTHOR]

Semi-algebraic proof systems such as sum-of-squares (SoS) have attracted a lot of attention due to their relation to approximation algorithms: constant degree semi-algebraic proofs lead to conjecturally optimal polynomial-time approximation algorithms for important NP -hard optimization problems. Motivated by the need to allow a more streamlined and uniform framework for working with SoS proofs than the restrictive propositional level, we initiate a systematic first-order logical investigation into the kinds of reasoning possible in algebraic and semi-algebraic proof systems. Specifically, we develop first-order theories that capture in a precise manner constant degree algebraic and semi-algebraic proof systems: every statement of a certain form that is provable in our theories translates into a family of constant degree polynomial calculus or SoS refutations, respectively; and using a reflection principle, the converse also holds. This places algebraic and semi-algebraic proof systems in the established framework of bounded arithmetic, while providing theories corresponding to systems that vary quite substantially from the usual propositional-logic ones. We give examples of how our semi-algebraic theory proves statements such as the pigeonhole principle, we provide a separation between algebraic and semi-algebraic theories, and we describe initial attempts to go beyond these theories by introducing extensions that use the inequality symbol, identifying along the way which extensions lead outside the scope of constant degree SoS. Moreover, we prove new results for propositional proofs, and specifically extend Berkholz's dynamic-by-static simulation of polynomial calculus (PC) by SoS to PC with the radical rule. [ABSTRACT FROM AUTHOR]

We will give a two-sort system which axiomatizes winning strategies for the combinatorial game Node Kayles. It is shown that our system captures alternating polynomial time reasonings in the sense that the provably total functions of the theory corresponds to those computable in APTIME. We will also show that our system is equivalently axiomatized by Sprague–Grundy theorem which states that any Node Kayles position is provably equivalent to some NIM heap. [ABSTRACT FROM AUTHOR]

We establish the compactly generated shape (H-shape) index theory for local semiflows on complete metric spaces via more general shape index pairs, which allows the phase space to be not separable. The main advantages are that the quotient space N/E is not necessarily metrizable for the shape index pair (N, E) and N \ E need not to be a neighbourhood of the compact invariant set. Moreover, we define H-shape cohomology groups and the H-shape cohomology index that is used to develop the Morse equations. Particularly, we apply H-shape index theory to an abstract retarded nonautonomous parabolic equation to illustrate these advantages for the new shape index theory, in which we obtain a special existence property of bounded full solutions. [ABSTRACT FROM AUTHOR]

In this paper, for any two elements y; u of a BCK-algebra X, we assign a subset of X, denoted by Sy(u), and investigate some related properties. We show that Sy(u) is a subalgebra of X for all y; u 2 X. Using these subalgebras, we characterize the involutive BCK-algebras, and give a necessary and sufficient condition for a bounded BCK-algebra to be a commutative BCK-chain. Finally, we show that the set of all subalgebras Sy(u) forms a bounded distributive lattice. [ABSTRACT FROM AUTHOR]

The |$\mathcal {H}$| -colouring problem for undirected simple graphs is a computational problem from a huge class of the constraint satisfaction problems (CSPs): an |$\mathcal {H}$| -colouring of a graph |$\mathcal {G}$| is just a homomorphism from |$\mathcal {G}$| to |$\mathcal {H}$| and the problem is to decide for fixed |$\mathcal {H}$|⁠ , given |$\mathcal {G}$|⁠ , if a homomorphism exists or not. The dichotomy theorem for the |$\mathcal {H}$| -colouring problem was proved by Hell and Nešetřil (1990, J. Comb. Theory Ser. B , 48, 92–110) (an analogous theorem for all CSPs was recently proved by Zhuk (2020, J. ACM , 67, 1–78) and Bulatov (2017, FOCS , 58, 319–330)), and it says that for each |$\mathcal {H}$|⁠ , the problem is either |$p$| -time decidable or |$NP$| -complete. Since negations of unsatisfiable instances of CSP can be expressed as propositional tautologies, it seems to be natural to investigate the proof complexity of CSP. We show that the decision algorithm in the |$p$| -time case of the |$\mathcal {H}$| -colouring problem can be formalized in a relatively weak theory and that the tautologies expressing the negative instances for such |$\mathcal {H}$| have polynomial proofs in propositional proof system |$R^*(log)$|⁠ , a mild extension of resolution. In fact, when the formulas are expressed as unsatisfiable sets of clauses, they have |$p$| -size resolution proofs. To establish this, we use a well-known connection between theories of bounded arithmetic and propositional proof systems. This upper bound follows also from a different construction in [ 1 ]. We complement this result by a lower bound result that holds for many weak proof systems for a special example of |$NP$| -complete case of the |$\mathcal {H}$| -colouring problem, using known results about the proof complexity of the pigeonhole principle. The main goal of our work is to start the development of some of the theories beyond the CSP dichotomy theorem in bounded arithmetic. We aim eventually—in a subsequent work—to formalize in such a theory the soundness of Zhuk's algorithm, extending the upper bound proved here from undirected simple graphs to the general case of directed graphs in some logical calculi. [ABSTRACT FROM AUTHOR]

We define an axiom schema IΔ0S for finite set theory with bounded induction on sets, analogous to the theory of bounded arithmetic, IΔ0, and use some of its basic model theory to establish some independence results for various axioms of set theory over IΔ0S. Then we ask: given a model M of IΔ0, is there a model of IΔ0S whose ordinal arithmetic is isomorphic to M? We show that the answer is yes if M⊧Exp. [ABSTRACT FROM AUTHOR]

In this paper, we give conditions under which the numerical range of a weighted composition operator, acting on a Hilbert space, contains zero as an interior point and we investigate extreme points of the numerical range of an operator acting on an arbitrary Banach space. Also, we give necessary and sufficient conditions under which the numerical range of an operator on some Banach spaces, to be closed. Finally, we characterize the structure of the numerical range of an operatoracting on Banach weighted Hardy spaces. [ABSTRACT FROM AUTHOR]

We show that ground state solutions to the nonlinear, fractional problem ( (−∆)su + V (x)u = f(x, u) in Ω, u = 0 in RN \ Ω, on a bounded domain Ω ⊂ RN , converge (along a subsequence) in L2 (Ω), under suitable conditions on f and V, to a solution of the local problem as s → 1−. [ABSTRACT FROM AUTHOR]

In this article, we present a detailed study of the logarithmic, iterated logarithmic and derived logarithmic densities of a bounded and positive arithmetic function and obtain as particular cases results on subset E of ℕ*, and give some applications to classical number theory. Some new existence criteria are established. This mathematical concept gives a prolongation of the results got in ([1], [3], [4], [5], [6], [7], [8]) [ABSTRACT FROM AUTHOR]

In this note we introduce generalised pairs from the perspective of the evolution of the notion of space in birational algebraic geometry. We describe some applications of generalised pairs in recent years and then mention a few open problems. [ABSTRACT FROM AUTHOR]

We study variants of Buss's theories of bounded arithmetic axiomatized by induction schemes disallowing the use of parameters, and closely related induction inference rules. We put particular emphasis on Π ^ i b induction schemes, which were so far neglected in the literature. We present inclusions and conservation results between the systems (including a witnessing theorem for T 2 i and S 2 i of a new form), results on numbers of instances of the axioms or rules, connections to reflection principles for quantified propositional calculi, and separations between the systems. [ABSTRACT FROM AUTHOR]

Abstract We deal with the algebraicity of an iterated Puiseux series in several variables in terms of the properties of its coefficients. Our aim is to generalize to several variables the results from [8]. We show that the algebraicity of such a series for given bounded degrees is determined by a finite number of explicit universal polynomial formulas. Conversely, given a vanishing polynomial, there is a closed-form formula for the coefficients of the series in terms of the coefficients of the polynomial and of a bounded initial part of the series. [ABSTRACT FROM AUTHOR]

Abstract We study the question of how quickly products of a fixed conjugacy class in the projective unitary group of a II 1 -factor von Neumann algebra cover the entire group. Our result is that the number of factors that are needed is essentially as small as permitted by the 1-norm – in analogy to results of Liebeck and Shalev for non-abelian finite simple groups. As an application of the techniques, we prove that every homomorphism from the projective unitary group of a finite factor to a Polish SIN group is continuous – a result which is even new for PU (n). Moreover, we show that the projective unitary group of a II 1 -factor carries a unique Polish group topology. [ABSTRACT FROM AUTHOR]

Let M be a smooth manifold with boundary and bounded geometry, ∂DM ⊂ ∂M be an open and closed subset of the boundary of M, P be a second order differential operator on M, and b be a first order differential operator on ∂M. Our operators act on sections of a vector bundle E → M with bounded geometry. We prove the regularity and well-posedness in the Sobolev spaces Hs (M; E), s ≥ 0, of the mixed Dirichlet-Robin boundary value problem P u = f in M, u = 0 on ∂DM, ∂P ν u + bu = 0 on ∂M \ ∂DM under the following four natural assumptions. First, we assume that P satisfies the strong Legendre condition and the first order terms are small. (In the scalar case, the strong Legendre condition reduces to the uniformly strong ellipticity condition.) Second, we assume that all the coefficients of P and all their covariant derivatives are bounded. Third, we assume that < b ≥ 0 and that there is > 0 and an open and closed subset ∂RM ⊂ ∂M \ ∂DM such that < b ≥ I on ∂RM. Finally, we assume that the distance to ∂RM ∪ ∂DM is uniformly bounded on M and that ∂RM ∪ ∂DM intersects all components of ∂M (i. e. (M, ∂RM ∪ ∂DM) has finite width). We include also some extensions of our main result in different directions. First, the finite width assumption is required for the Poincar'e inequality on manifolds with bounded geometry, a result for which we give a new, more general proof. Second, we consider also the case when we have a decomposition of the vector bundle E (instead of a decomposition of the boundary). Third, we also consider operators with non-smooth coefficients, but, in this case, we need to limit the range of s. Finally, we also consider the case of uniformly strongly elliptic operators and discuss the equivalence between the uniform Agmon condition and the G˚arding inequality. The main novelty of our results is that they are formulated on a non-compact manifold. [ABSTRACT FROM AUTHOR]

Two years ago, during the 21st European Conference on Iteration Theory, Gregory Derfel asked: "Does there exist a non-trivial bounded continuous solution of the equation 2f(x)=f(x-1)+f(-2x)?" He repeated the question during the 55th International Symposium on Functional Equations. In this paper we present a partial solution of a more general problem, connected to the functional equation f(x)=M(f(x+t1),f(x+t2),...,f(x+tn-1),f(ax)), where n∈N,t1,t2,...,tn-1∈R\{0},a∈(-∞,0) and M is a given function in n variables satisfying some additional properties. In particular, M can be a weighted quasi-arithmetic mean in n variables. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with an attraction–repulsion chemotaxis–(Navier)–Stokesmodel with logistic source n t + u ⋅ ∇ n = Δ n − χ ∇ ⋅ (n ∇ c) + ξ ∇ ⋅ (n ∇ v) + μ n (1 − n) , (x , t) ∈ Ω × (0 , ∞) , c t + u ⋅ ∇ c = Δ c − c + n , (x , t) ∈ Ω × (0 , ∞) , v t + u ⋅ ∇ v = Δ v − v + n , (x , t) ∈ Ω × (0 , ∞) , u t + κ (u ⋅ ∇) u = Δ u + ∇ P + n ∇ ϕ , (x , t) ∈ Ω × (0 , ∞) , ∇ ⋅ u = 0 , (x , t) ∈ Ω × (0 , ∞) , under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R N , N = 2 , 3 , where κ ∈ { 0 , 1 } , the parameters χ , ξ and μ are positive. This system describes the evolution of cells which react on two different chemical signals in a liquid surrounding environment. The cells and chemical substances are transported by a viscous incompressible fluid under the influence of a force due to the aggregation of cells. Firstly, when N = 2 and κ = 1 , based on the standard heat-semigroup argument, it is proved that for all appropriately regular nonnegative initial data and any positive parameters, this system possesses a unique global bounded solution. Secondly, when N = 3 and κ = 0 , by using the maximal Sobolev regularity and semigroup technique, it is proved that the system admits a unique globally bounded classical solution provided that there exists θ 0 > 0 such that χ + ξ μ < θ 0. Finally, by means of energy functionals, it is shown that the global bounded solution of the above system converges to the constant steady state. Furthermore, we give the precise convergence rates. [ABSTRACT FROM AUTHOR]

Abstract We consider the following chemotaxis–Stokessystem with rotation n t = Δ n − ∇ ⋅ (n S (x , n , c) ⋅ ∇ c) − u ⋅ ∇ n , c t = Δ c − f (x , n , c) − u ⋅ ∇ c , u t = Δ u + ∇ P + n ∇ ϕ , ∇ ⋅ u = 0 in Ω × (0 , T) , subject to the non-flux boundary conditions for n and c , as well as the Dirichlet boundary condition for u , where the bounded smooth domain Ω ⊂ R 3 , the matrix-valued function S ∈ C 2 (Ω ̄ × [ 0 , ∞) 2 ; R 3 × 3) fulfills | S (x , n , c) | ≤ S 0 (c) (1 + n) θ for all (x , n , c) ∈ Ω ̄ × [ 0 , ∞) 2 with S 0 nondecreasing, and f ∈ C 1 (Ω ̄ × [ 0 , ∞) 2 ; R) satisfies 0 ≤ f (x , n , c) ≤ f 0 (c) (n + 1) with f 0 nondecreasing and f (x , n , 0) = 0. It was proved that for any θ > 0 , the initial–boundary value problem possesses a unique globally bounded classical solution. [ABSTRACT FROM AUTHOR]

Summary: For the global stabilization of a family of feedforward nonlinear time‐delay systems whose linearized systems consist of a chain of identical oscillators, a saturated feedback control is established based on a special canonical form. The proposed control laws use not only the current states but also the delayed states for feedback, which helps maintain the decoupling property in the recursive design. Moreover, explicit conditions guaranteeing the stability of the closed‐loop system are provided. When the nonlinear terms vanish and even the oscillators are distinct, a modified saturated feedback control can still be established for the corresponding global stabilization problem. Two numerical examples are given to illustrate the effectiveness of the proposed approaches. [ABSTRACT FROM AUTHOR]

Abstract The following result gives the flavor of this paper: Let t , k and q be integers such that q ≥ 0 , 0 ≤ t < k and t ≡ k (mod 2) , and let s ∈ [ 0 , t + 1 ] be the unique integer satisfying s ≡ q + k − t − 2 2 (mod (t + 2)). Then for any integer n such that n ≥ max ⁡ { k , 1 2 (t + 2) k 2 + q − s t + 2 k − t 2 + s } and any function f : [ n ] → { − 1 , 1 } with | ∑ i = 1 n f (i) | ≤ q , there is a set B ⊆ [ n ] of k consecutive integers with | ∑ y ∈ B f (y) | ≤ t. Moreover, this bound is sharp for all the parameters involved and a characterization of the extremal sequences is given. This and other similar results involving different subsequences are presented, including decompositions of sequences into subsequences of bounded weight. [ABSTRACT FROM AUTHOR]

Abstract For the Monge–Ampère equations det D 2 u = f (u) , we find auxiliary curvature functions which attain respective maximum on the boundary under some conditions for f. Moreover, we obtain the upper bounded estimates for the Gauss curvature and mean curvature of the level sets for the solution to this equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the boundedness of the higher order Riesz transforms R, R* and their commutators [b, R], [b, R*] on local generalized Morrey spaces LMp,φα,V,{x0} and vanishing generalized Morrey spaces VMp,φα,V related to Schrödinger type operator. We find the sufficient conditions on the pair (φ1, φ2) which ensures the boundedness of these operators from one local generalized Morrey space LMp,φα,V,{x0} to another LMp,φ2α,V,{x0} and from one vanishing generalized Morrey space VMp,φα,V to another VMp,φ2α,V. [ABSTRACT FROM AUTHOR]

E(1) μk0,E(2) μkM, E(3) μk0;E(4) μK1 and E(5) μkM of any function f on [0; 1) having bounded derivative are obtained by extended Legendre wavelet method. These new estimators are sharper and best possible in wavelet analysis. [ABSTRACT FROM AUTHOR]

Abstract Recent papers investigated the maximum infection times t P 3 (G) , t g d (G) and t m o (G) of the P 3 convexity, geodesic convexity and monophonic convexity, respectively. In Benevides et al. (2016) and Costa et al. (2015), it was proved that deciding whether t g d (G) ≥ 2 or t m o (G) ≥ 2 are NP-Complete problems even for bipartite graphs. In Marcilon et al. (2014), it was proved that, in bipartite graphs, deciding whether t P 3 (G) ≥ k is polynomial time solvable for k ≤ 4 , but is NP-Complete for k ≥ 5. In this paper, we prove that, in grid graphs with maximum degree 3, t P 3 (G) is NP-hard (answering a question of Benevides et al. (2015)), but is polynomial time solvable if the grid graph is solid. Moreover, we prove that deciding whether t P 3 (G) ≥ n − k is polynomial time solvable for any fixed k , but is NP-Complete for k = n ε − 4 for every fixed 0 < ε ≤ 1 , generalizing the main result of Benevides et al. (2015). Finally, for any fixed Δ , we prove that, in graphs with bounded maximum degree Δ , deciding whether t P 3 (G) ≥ k = Θ (log n) is polynomial time solvable if Δ ≤ 3 , but is NP-Complete if Δ ≥ 4. [ABSTRACT FROM AUTHOR]

Abstract A bounded-degree-1 set S in an undirected graph G = (V , E) is a vertex subset such that the maximum degree of G [ S ] is at most one. Given a graph G , the Maximum Bounded-Degree-1Set (Max 1-bds) problem is to find a bounded-degree-1 set S of maximum size in G.A notion related to bounded-degree sets is that of an s - plex used to define the cohesiveness of subgraphs in social networks. An s -plex S in a graph G = (V , E) is a vertex subset such that for each v ∈ S , deg G [ S ] (v) ≥ | S | − s. One can easily show that a graph G has a2-plex of size k iff the complement graph of G has a bounded-degree-1 set of size k. Both the Maximum 2-Plex problem and the Maximum Bounded-Degree-1 Set problem are NP-hard. We give a simple branch-and-reduce algorithm running in polynomial space and applying branching strategies with at most three branches for Max 1-bds. We analyze the running time of the algorithm using measure-and-conquer and show that it runs in time O ∗ (1. 461 3 n) which is faster than previous exact algorithms. Moreover, we show that the Max 1-bds problem cannot be approximated to a ratio greater than n ϵ − 1 in polynomial time for all ϵ > 0 unless P = NP. Some moderately exponential time approximation algorithms are given for Max 1-bds and its dual problem. [ABSTRACT FROM AUTHOR]

Abstract The Defensive Alliance problem has been studied extensively during the last fifteen years, but the question whether it is FPT when parameterized by treewidth has still remained open. We show that this problem is W [ 1 ] -hard. This puts it among the few problems that are FPT when parameterized by solution size but not when parameterized by treewidth (unless FPT = W [ 1 ]). [ABSTRACT FROM AUTHOR]

We link two concepts from the literature, namely hard sequences for the satisfiability problem sat and so‐called pseudo proof systems proposed for study by Krajíček. Pseudo proof systems are elements of a particular nonstandard model constructed by forcing with random variables. We show that the existence of mad pseudo proof systems is equivalent to the existence of a randomized polynomial time procedure with a highly restrictive use of randomness which produces satisfiable formulas whose satisfying assignments are probably hard to find. [ABSTRACT FROM AUTHOR]

We prove the equivalence of Sarnak's conjecture on Möbius orthogonality with a Kolmogorov type property conjectured by Veech for Furstenberg systems of the Möbius function. This yields a combinatorial condition on the Möbius function itself which is equivalent to Sarnak's conjecture. As a matter of fact, our arguments remain valid in a larger context: we characterize all bounded arithmetic functions orthogonal to all topological systems whose all ergodic measures yield systems from a fixed characteristic class (zero entropy class is an example of such a characteristic class) with the characterization persisting in the logarithmic setup. As a corollary, we obtain that the logarithmic Sarnak's conjecture holds if and only if the logarithmic Möbius orthogonality is satisfied for all dynamical systems whose ergodic measures yield nilsystems. [ABSTRACT FROM AUTHOR]

Abstract We prove partial regularity of stationary solutions and minimizers u from a set Ω ⊂ R n to a Riemannian manifold N , for the functional ∫ Ω F (x , u , | ∇ u | 2) d x. The integrand F is convex and satisfies some ellipticity and boundedness assumptions. We also develop a new monotonicity formula and an ϵ -regularity theorem for such stationary solutions with no restriction on their images. We then use the idea of quantitative stratification to show that the k-th strata of the singular set of such solutions are k-rectifiable. [ABSTRACT FROM AUTHOR]

Abstract We provide a unified approach to Fraïssé limits in functional analysis, including the Gurarij space, the Poulsen simplex, and their noncommutative analogs. We obtain in this general framework many known and new results about the Gurarij space and the Poulsen simplex, and at the same time establish their noncommutative analogs. Particularly, we construct noncommutative analogs of universal operators in the sense of Rota. [ABSTRACT FROM AUTHOR]

In the present paper, we carry forward R. Rochberg's work [4] on the boundedness of composition operators associated to continuous symbols of norm greater than 1. More general sufficient conditions and necessary conditions are given for the boundedness of K φ : = P H 2 D φ where D φ : C [ z ] → L 2 ( T ) is the composition operator of the symbol φ . [ABSTRACT FROM AUTHOR]

In this paper we characterize the boundedness and compactness of the weighted composition operator from the analytic Morrey spaces L2;λ to the Zygmund space Z, and the little analytic Morrey spaces L2λ0 to the little Zygmund space Z0, respectively. [ABSTRACT FROM AUTHOR]

Abstract: For a reduced free product C ∗‐algebra ( A , φ ) = ( A 1 , φ 1 ) ★ ( A 2 , φ 2 ), we prove a boundary rigidity result for the embedding of A into its associated C ∗‐algebra Δ T ( A , φ ) under the assumption that the GNS representations associated with φ 1 and φ 2 are essential and A 1 satisfies a suitable diffuseness condition. This provides new examples of rigid embeddings of exact C ∗‐algebras into purely infinite simple nuclear C ∗‐algebras. [ABSTRACT FROM AUTHOR]

Abstract: In this note we show a comparison principle for nonlinear heat Rockland operators on graded groups. We give a simple proof for it using purely algebraic relations. As an application of the established comparison principle we prove the global in t‐boundedness of solutions for a class of nonlinear equations for the heat p‐sub‐Laplacian on stratified groups. [ABSTRACT FROM AUTHOR]